How to Make It
Preheat oven to 350°. Coat a 9-inch round cake pan with cooking spray. Line bottom of pan with parchment paper; lightly coat paper with cooking spray.
To prepare topping, melt 3 tablespoons butter in a small saucepan over medium heat; add 1/3 cup brown sugar to pan, stirring with a whisk until sugar dissolves. Pour mixture into prepared pan, spreading evenly with a silicone spatula. Grate rind from 1 blood orange and grapefruit; set aside. Peel 1 blood orange and grapefruit; cut each crosswise into thin slices (you should have 5 to 6 slices of each). Carefully remove seeds from fruit. Arrange citrus rounds over sugar mixture in pan.
To prepare cake, weigh or lightly spoon flours into dry measuring cups; level with a knife. Combine flours, baking powder, cardamom, salt, and baking soda in a bowl, stirring with a whisk; set aside.
Combine granulated sugar, 1/3 cup brown sugar, 1/4 cup butter, and oil in the bowl of a stand mixer fitted with the paddle attachment; beat at medium speed 4 to 5 minutes, scraping down sides of bowl as necessary. Add eggs, 1 at a time, beating well after each addition. Beat in citrus rinds and 1 teaspoon vanilla.
Combine 1/3 cup sour cream, milk, wine, and blood orange juice (from remaining 1 orange) in a small bowl, stirring with a whisk until smooth. Add flour mixture and sour cream mixture alternately to butter mixture, beginning and ending with flour mixture, beating just until combined. Pour batter over fruit in pan, smoothing with a spatula. Bake at 350° for 55 minutes or until a wooden pick inserted in center comes out clean. Cool in pan 20 minutes on a wire rack. Loosen cake from sides of pan using a knife. Place a serving plate upside down on top of cake; carefully invert pan onto plate. Let stand 5 minutes or until cake releases from pan; carefully remove pan and parchment paper. Sprinkle cake evenly with thyme.
To prepare sour whipped cream, combine 1/2 cup sour cream and remaining ingredients in a chilled metal bowl; beat with a whisk until mixture thickens. Serve with cake.